THE ARITHMETIC OF POLYNOMIAL DYNAMICAL PAIRS

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THE ARITHMETIC OF POLYNOMIAL DYNAMICAL PAIRS

Charles Favre, Thomas Gauthier

To cite this version:

Charles Favre, Thomas Gauthier. THE ARITHMETIC OF POLYNOMIAL DYNAMICAL PAIRS.

2020. �hal-02556315�

Charles Favre Thomas Gauthier

THE ARITHMETIC OF

POLYNOMIAL DYNAMICAL

PAIRS

Charles Favre

CMLS, ´ Ecole polytechnique, CNRS, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France.

PIMS, University of British Columbia, Department of Mathematics, Vancouver, BC, V6T 1Z2, Canada.

E-mail : charles.favre@polytechnique.edu Thomas Gauthier

CMLS, ´ Ecole polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France.

E-mail : thomas.gauthier@polytechnique.edu

The second author is partially supported by ANR project “Fatou” ANR-17-

CE40-0002-01.

Aux math´ ematiciens qui nous ont tant inspir´ es Adrien Douady,

Tan Lei, Jean-Christophe Yoccoz.

THE ARITHMETIC OF POLYNOMIAL DYNAMICAL PAIRS

Charles Favre, Thomas Gauthier

Abstract. — We study one-dimensional algebraic families of pairs given by a polynomial with a marked point. We prove an ”unlikely intersection” state- ment for such pairs thereby exhibiting strong rigidity features for these pairs.

We infer from this result the dynamical Andr´ e-Oort conjecture for curves in the moduli space of polynomials, by describing one-dimensional families in this parameter space containing infinitely many post-critically finite parameters.

R´ esum´ e (Etude arithm´ etique des paires dynamiques polynomiales) Nous ´ etudions les familles alg´ ebriques param´ etr´ ees par une courbe des paires de polynˆ omes munis d’un point marqu´ e. Nous d´ emontrons un ´ enonc´ e

”d’intersection improbable” pour ces paires, qui se traduit par une forme forte

de rigidit´ e pour ces paires. Nous en d´ eduisons la conjecture d’Andr´ e-Oort

dynamique pour les courbes dans l’espace des modules des polynˆ omes, en

d´ ecrivant les courbes de cet espace contenant une infinit´ e de param` etres

post-critiquement finis.

CONTENTS

Introduction. . . . 1

Notations. . . 15

1. Geometric background. . . 19

1.1. Analytic geometry. . . 19

1.2. Potential theory. . . 25

1.3. Line bundles on curves. . . 32

1.4. Adelic metric, Arakelov heights and equidistribution . . . 34

1.5. The parameter space of polynomials. . . 38

1.6. Adelic series and Xie’s algebraization theorem . . . 40

2. Polynomial dynamics. . . 45

2.1. Fatou-Julia theory. . . 45

2.2. Green functions and equilibrium measure. . . 49

2.3. Examples. . . 53

2.4. B¨ ottcher coordinates & Green functions . . . 56

2.5. Polynomial dynamics over a global field. . . 62

2.6. Bifurcations in holomorphic dynamics . . . 64

2.7. Components of preperiodic points . . . 66

3. Dynamical symmetries. . . 71

3.1. The group of dynamical symmetries of a polynomial . . . 72

3.2. Symmetry groups in family. . . 75

3.3. Algebraic characterization of the group of dynamical symmetries . . 77

3.4. Primitive families of polynomials . . . 80

3.5. Ritt’s theory of composite polynomials . . . 84

3.6. Stratification of the parameter space in low degree. . . 94

3.7. Open problems. . . 96

4. Polynomial dynamical pairs. . . 99

4.1. Holomorphic dynamical pairs and proof of Theorem A . . . 100

viii

CONTENTS

4.2. Algebraic dynamical pairs. . . 112

4.3. Family of polynomials and Green functions . . . 119

4.4. Arithmetic polynomial dynamical pairs. . . 120

5. Entanglement of dynamical pairs. . . 127

5.1. Dynamical entanglement . . . 127

5.2. Dynamical pairs with identical measures. . . 131

5.3. Multiplicative dependence of the degrees. . . 135

5.4. Proof of the implication (2) ⇒ (3) of Theorem B . . . 139

5.5. Proof of Theorem C. . . 149

5.6. Further results and open problems. . . 154

6. Entanglement of marked points. . . 159

6.1. Proof of Theorem D. . . 159

6.2. Proof of Theorem E. . . 161

7. The unicritical family. . . 167

7.1. General facts. . . 167

7.2. Unlikely intersection in the unicritical family . . . 170

7.3. Archimedean rigidity. . . 171

7.4. Connectedness of the bifurcation locus. . . 172

7.5. Some experiments. . . 173

8. Special curves in the parameter space of polynomials. . . 177

8.1. Special curves in the moduli space of polynomials. . . 178

8.2. Marked dynamical graphs. . . 180

8.3. Dynamical graphs attached to special curves. . . 185

8.4. Realization theorem. . . 187

8.5. Special curves and special critically marked dynamical graphs. . . 208

8.6. Realizability of PCF maps. . . 212

8.7. Special curves in low degrees . . . 220

8.8. Open questions on the geometry of special curves. . . 223

Index. . . 225

Bibliography. . . 227

INTRODUCTION

This book is intented as an exploration of the moduli space Poly

of complex polynomials of degree d ≥ 2 in one variable using tools primarily coming from arithmetic geometry.

The Mandelbrot set in Poly

has undoubtedly been the focus of the most comprehensive set of studies, and its local geometry is still an active research field in connection with the Fatou conjecture, see [19] and the references therein. In their seminal work, Branner and Hubbard [29, 30] gave a topo- logical description of the space of cubic polynomials with disconnected Julia sets using combinatorial tools. In any degree, Poly

is a complex orbifold of dimension d − 1, and is therefore naturally amenable to complex analysis and in particular to pluripotential theory. This observation has been particularly fruitful to describe the locus of unstability, and to investigate the boundary of the connectedness locus. DeMarco [42] constructed a positive closed (1, 1) current whose support is precisely the set of unstable parameters. Dujardin and the first author [59] then noticed that the Monge-Amp` ere measure of this current defines a probability measure µ

whose support is in a way the right generalization of the Mandelbrot set in higher degree, capturing the part of the moduli space where the dynamics is the most unstable (see also [11] for the case of rational maps). The support of µ

has a very intricate structure:

it was proved by Shishikura [132] in degree 2 and later generalized in higher degree by the second author [74] that the Hausdorff dimension of the support of µ

was maximal equal to 2(d − 1).

A polynomial is said to be post-critically finite (or PCF) if all its critical points have a finite orbit. The Julia set of a PCF polynomial is connected, of measure zero, and the dynamics on it is hyperbolic off the post-critical set.

PCF polynomials form a countable subset of larger classes of polynomials (such

as Misiurewicz, or Collet-Eckmann) for which the thermodynamical formalism

2

INTRODUCTION

is well understood, [121, 122]. They also play a pivotal role in the study of the connectedness locus of Poly

: their distribution was described in a series of papers [65, 76, 77] and proved to represent the bifurcation measure µ

.

PCF polynomials are naturally defined by d − 1 equations of the form P

(c) = P

(c) where c denotes a critical point and n, m are two distinct integers. In the moduli space, these equations are algebraic with integral co- efficients, so that any PCF polynomial is in fact defined over a number field.

Ingram [94] has pushed this remark further and has built a natural height h

: Poly

( ¯ Q ) → R

for which the set of PCF polynomials coincides with {h

= 0}.

Height theory yields interesting new perspectives on the geometry of Poly

, and more specifically on the distribution of PCF polynomials. We will be mostly interested here in the so-called dynamical Andr´ e-Oort conjecture which appeared in [6], see also [136].

This remarkable conjecture was set out by Baker and DeMarco who were mo- tivated by deep analogies between PCF dynamics and CM points in Shimura varieties, and more specifically by works by Masser-Zannier [26, 105, 149]

on torsion points in elliptic curves. An historical account of the introduction of these ideas in arithmetic dynamics is given in [6, §1.2], and [5, §1.2], see also [78]. We note that this analogy is going far beyond the problems consid- ered in this book, and applies to various conjectures described in [135, 45].

We refer to the book by Zannier [149] for a beautiful discussion of unlikely intersection problems in arithmetic geometry.

Baker and DeMarco proposed to characterize irreducible subvarieties of Poly

(or more generally of the moduli space of rational maps) containing a Zariski dense subset of PCF polynomials, and conjectured that such vari- eties were defined by critical relations. This conjecture was proven in degree 3 in [66] and [88], and for unicritical polynomials in [82] and [83].

It is our aim to give a proof of that conjecture for curves in Poly

for any d ≥ 2, and based on this result to attempt a classification of these curves in terms of combinatorial data encoding critical relations.

Our proof roughly follows the line of arguments devised in the original pa-

per of Baker and DeMarco, and relies on equidistribution theorems of points

of small height by Thuillier [140] and Yuan [147]; on the expansion of the

B¨ ottcher coordinates; and on Ritt’s theory characterizing equalities of compo-

sition of polynomials.

INTRODUCTION

We needed, though, to overcome several important technical difficulties, such as proving the continuity of metrics naturally attached to families of polyno- mials. We also had to inject new ingredients, most notably some dynamical rigidity results concerning families of polynomials with a marked point whose bifurcation locus is real-analytic.

For the most part of the memoir, we shall work in the general context of polynomial dynamical pairs (P, a) parameterized by a complex affine curve C, postponing the proof of the dynamical Andr´ e-Oort conjecture to the last chapter. We investigate quite generally the problem of unlikely intersection that was promoted in the context of torsion points on elliptic curves by Zannier and his co-authors [105, 149], and later studied by Baker and DeMarco [5, 6]

in our context. This problem amounts to understanding when two polynomial dynamical pairs (P, a) and (Q, b) parameterized by the same curve C have an infinite set of common parameters for which the marked points are preperiodic.

We obtain quite definite answers for polynomial pairs, and we prove finiteness theorems that we feel are of some interest for further exploration.

We have tried to review all the necessary material for the proof of the dy- namical Andr´ e-Oort conjecture, but we have omitted some technical proofs that are already available in the literature in an optimal form. On the other hand, we have made some efforts to clarify some proofs which we felt too sketchy in the literature. The group of dynamical symmetries of a polynomial play a very important role in unlikely intersection problems, and we have thus included a detailed discussion of this notion.

Let us now describe in more detail the content of the book.

Polynomial dynamical pairs. — In this paragraph we present the main players of our memoir. The central notion is the one of polynomial dy- namical pair parameterized by a curve. Such a pair (P, a) is by definition an algebraic family of polynomials P

parameterized by an irreducible affine curve C defined over a field K, accompanied by a regular function a ∈ K[C]

which defines an algebraically varying marked point. Most of the time, our objects will be defined over the field of complex numbers K = C , but it will also be important to consider dynamical pairs over other fields like number fields, p-adic fields, or finite fields.

Any polynomial dynamical pair leaves a ”trace” on the parameter space C,

which may take different forms. Suppose first that K is an arbitrary field, and

let ¯ K be an algebraic closure of K. The first basic object to consider is the set

4

INTRODUCTION

Preper(P, a) of (closed) points t ∈ C( ¯ K) such that a(t) is preperiodic under P

. This set is either equal to C or is at most countable.

A slightly more complicated but equally important object one can attach to (P, a) is the following divisor. Let ¯ C be the completion of C, that is the unique projective algebraic curve containing C as a Zariski dense open subset, and smooth at all points ¯ C \ C. Points in ¯ C \ C are called branches at infinity of C. Any pair (P, a) induces an effective divisor D

on ¯ C, which is obtained by setting

(1) ord

(D

) := lim

n→∞

− 1

d

min{0, ord

(P

(a))},

for any branch c at infinity. The limit is known to exist and is always a rational number, see §4.2.2.

When K = C , one can associate more topological objects to a dynamical pair. One can consider the locus of stability of the pair (P, a) which consists of the open set over which the family of holomorphic maps {P

(a)}

is normal.

Its complement is the bifurcation locus which we denote by Bif(P, a). This set can be characterized using potential theory as follows. Recall the definition of the Green function of a polynomial P of degree d:

g

(z) := lim

n→∞

1

d

max{log |P

(z)|, 0},

so that {g

= 0} is the filled-in Julia set of P consisting of those points having bounded orbits. On the parameter space C, we then define the function

g

(t) = g

(a(t)).

It is a non-negative continuous subharmonic function on C, and the support of the measure µ

= ∆g

is precisely equal to Bif(P, a). Of crucial technical importance is the following result from [67] which relates the function g

to the divisor defined above.

Theorem 1. — In a neighborhood of any branch at infinity c ∈ C, one has ¯ the expansion

g

(t) = ord

(D

) log |t|

+ ˜ g(t)

where t is a local parameter centered at c and g ˜ is continuous at 0.

This result can be interpreted in the langage of complex geometry by saying that g

induces a continuous semi-positive metrization on the Q -line bundle O

(D

). This technical fact will be of crucial importance when we shall apply techniques from arithmetic geometry.

Let us now suppose that K = K is a number field. For any place v of K ,

denote by K

the completion of K , and by C

the completion of its algebraic

INTRODUCTION

closure. It is then possible to mimic the previous constructions at any (finite or infinite) place v of K so as to obtain functions g

: C

→ R

on the ana- lytification (in the sense of Berkovich) C

of the curve C over C

. Combining these functions yield a height function h

: C( ¯ K ) → R

.

Alternatively, we may start from the standard Weil height h

: P

( ¯ K ) → R

, see e.g. [90]. Then for any polynomial with algebraic coefficients, we define its canonical height [35]:

h

(z) := lim

n→∞

1

d

h

(P

(z)),

and finally set h

(t) := h

(a(t)). Using the Northcott theorem, one obtains that {h

= 0} coincides with the set Preper(P, a) of parameters t ∈ C( ¯ K ) for which a(t) is a preperiodic point of P

.

It is an amazing fact that all the objects attached to a dynamical pair (P, a) we have seen so far are tightly interrelated, as the next theorem due to DeMarco [44] shows.

An isotrivial pair (P, a) is a pair which is conjugated to a constant poly- nomial and a constant marked point possibly after a base change. A marked point is stably preperiodic when there exist two integers n > m such that P

(a(t)) = P

(a(t)).

Theorem 2. — Let (P, a) be a dynamical pair of degree d ≥ 2 parametrized by an affine irreducible curve C defined over a number field K . If the pair is not isotrivial, then the following assertions are equivalent:

1. the set Preper(P, a) is equal to C( ¯ K );

2. the marked point is stably preperiodic;

3. the divisor D

of the pair (P, a) vanishes;

4. for any Archimedean place v, the bifurcation measure µ

:= ∆g

vanishes;

5. the height h

is identically zero.

A pair (P, a) which satisfies either one of the previous conditions is said to be passive, otherwise it is called an active pair . For an active pair, Preper(P, a) is countable, the bifurcation measure µ

is non trivial, and the height h

is non zero.

Holomorphic rigidity for dynamical pairs. — Rigidity results are per-

vasive in (holomorphic) dynamics. One of the most famous rigidity result was

obtained by Zdunik [150] and states the following. The measure of max-

imal entropy of a polynomial P is absolutely continuous to the Hausdorff

6

INTRODUCTION

measure of its Julia set iff P is conjugated by an affine transformation to either a monomial map M

(z) = z

, or to a Chebyshev polynomial ±T

where T

(z + z

) = z

+ z

. In particular, these two families of examples are the only ones having a smooth Julia set, a theorem due to Fatou [63].

The following analog of Zdunik’s result for polynomial dynamical pairs is our first main result.

Theorem A. — Let (P, a) be a dynamical pair of degree d ≥ 2 parametrized by a connected Riemann surface C. Assume that Bif(P, a) is non-empty and included in a smooth real curve. Then one of the following holds:

– either P

is conjugated to M

or ±T

for all t ∈ C;

– or there exists a univalent map ı: D → C such that ı

(Bif(P, a)) is a non-empty closed and totally disconnected perfect subset of the real line and the pair (P ◦ ı, a ◦ ı) is conjugated to a real family over D .

We say that a polynomial dynamical pair (P, a) parameterized by the unit disk is a real family whenever the power series defining the coefficients of P and the marked point have all real coefficients.

The previous theorem is a crucial ingredient for handling the unlikely in- tersection problem that we will describe later. Its proof builds on a transfer principle from the parameter space to the dynamical plane which can be de- composed into two parts.

The first step is to find a parameter t

at which a(t

) is preperiodic to a repelling orbit of P

and such that t 7→ a(t) is transversal at t

to the preperi- odic orbit degenerating to a(t

). This step builds on an argument of Dujardin [58]. The second step relies on Tan Lei’s similarity theorem [139] which shows that the bifurcation locus Bif(P, a) near t

is conformally equivalent at small scales to the Julia set of P

.

Combining these two ingredients, we see that if Bif(P, a) is connected, then Zdunik’s theorem implies that P

is isotrivial conjugated to M

or ±T

for all t ∈ C. When Bif(P, a) is disconnected, then we prove that all multipliers of P

are real and we conclude that P

is real for all nearby parameters using an argument of Eremenko and Van Strien [62].

In many results that we present below, we shall exclude all polynomials

that are affinely conjugated to either M

or ±T

. These dynamical systems

carry different names in the literature: Zdunik [150] name them maps with

parabolic orbifolds; they are called special in [48, 117]; and Medvedev and

Scanlon call them non-disintegrated polynomials, see the discussion on [108,

p.16]. We shall refer them to as integrable polynomials by analogy with the

INTRODUCTION

notion of integrable systems in hamiltonian dynamics (see [38, 143]). A family of polynomials {P

}

will be called non-integrable whenever there exists a dense open set U ⊂ C such that P

is not integrable for any t ∈ U .

Unlikely intersections for polynomial dynamical pairs. — Our next objective is to investigate the problem of characterizing when two dynamical pairs (P, a) and (Q, b) parameterized by the same algebraic curve C leave the same ”trace” on C.

Analogies with arithmetic geometry suggested that the quite weak condition of Preper(P, a)∩Preper(Q, b) being infinite in fact implies very strong relations between the two pairs. This phenomenon was first observed for Latt` es maps by Masser and Zannier [105], and later for unicritical polynomials by Baker and DeMarco [5], and for more general families of polynomials parameterized by the affine line by Ghioca, Hsia and Tucker [80]. We refer to the surveys [78], [45] and [14] where this problem is also addressed.

A precise conjecture was formulated by DeMarco in [46, Conjecture 4.8]:

up to symmetries and taking iterates the two families P and Q are actually equal, and the marked points belong to the same grand orbit. In other words, the existence of unlikely intersections forces some algebraic rigidity between the dynamical pairs.

We prove here DeMarco’s conjecture for polynomial dynamical pairs defined over a number field.

Theorem B. — Let (P, a) and (Q, b) be active non-integrable dynamical pairs parametrized by an irreducible algebraic curve C of respective degree d, δ ≥ 2. Assume that the two pairs are defined over a number field K . Then, the following are equivalent:

1. the set Preper(P, a) ∩ Preper(Q, b) is an infinite subset of C( ¯ K );

2. the two height functions h

, h

: C( ¯ K ) → R

are proportional;

3. there exist integers N, M ≥ 1, r, s ≥ 0, and families R, τ and π of poly- nomials of degree ≥ 1 parametrized by C such that

(†) τ ◦ P

= R ◦ τ and π ◦ Q

= R ◦ π, and τ(P

(a)) = π(Q

(b)).

It is not difficult to see that (3)⇒(2)⇒(1) so that the main content of the theorem is the implication (1)⇒(3). To obtain (1)⇒(2), we first apply Yuan-Thuillier’s equidistribution result [140, 147] of points of small height:

it is precisely at this step that the continuity of ˜ g in Theorem 1 is crucial.

8

INTRODUCTION

This allows one to prove that the bifurcation measures µ

and µ

are proportional at any place v of K . From there, one infers the proportionality of height functions i.e. (2) using our above rigidity result (Theorem A).

The implication (2)⇒(3) is more involved. We first prove that deg(P ) and deg(Q) are multiplicatively dependent using an argument taken from [60]

which consists of computing the H¨ older constants of continuity of the po- tentials of the bifurcation measures at a complex place. From this, we obtain (3) by combining in a quite subtle way several ingredients including:

– a precise understanding of the expansion at infinity of the B¨ ottcher coor- dinate;

– an algebraization result of germs of curves defined by adelic series due to Xie [144];

– and the classification of invariant curves by product maps (z, w) 7→

(R(z), R(w)).

The latter result is due to Medvedev and Scanlon [108] whose proof elabo- rates on Ritt’s theory [124]. This theory aims at describing all possible ways a polynomial can be written as the composition of lower degree polynomials.

It is very combinatorial in nature and was treated by several authors, by Zan- nier [148], by M¨ uller-Zieve [153], see also the references therein. Of particular relevance for us are the series of papers by Pakovich [115, 116, 117], and by Ghioca, Nguyen and their co-authors [84, 86].

As mentioned above, the line of arguments for proving Theorem B is mostly taken from the seminal paper of Baker and DeMarco, but with considerably more technical issues. The core of the proof takes about 8 pages and is the content of §5.4.

It would be desirable to extend Theorem B to families defined over an ar- bitrary field of characteristic zero. Reducing to the case over a number field typically uses a specialization argument. We faced an essential difficulty in the course of this argument, and thus had to require an additional assumption.

Theorem C. — Pick any irreducible algebraic curve C defined over a field of characteristic 0. Let (P, a) and (Q, b) be active non-integrable dynamical pairs parametrized by C of respective degree d, δ ≥ 2. Assume that

( M ) any branch at infinity c of C belongs to the support of the divisor D

. Then, the following are equivalent:

1. the set Preper(P, a) ∩ Preper(Q, b) is an infinite subset of C;

INTRODUCTION

2. there exist integers N, M ≥ 1, r, s ≥ 0, and families R, τ and π of poly- nomials parametrized by C such that

τ ◦ P

= R ◦ τ and π ◦ Q

= R ◦ π, and τ(P

(a)) = π(Q

(b)).

Note that although ( M ) may not hold in general, it is always satisfied when C admits a unique branch at infinity, e.g. when C is the affine line. In particular, our result yields a far-reaching generalization of [5, Theorem 1.1].

In the sequel, we call two active dynamical pairs (P, a) and (Q, b) entan- gled when Preper(P, a) ∩ Preper(Q, b) is infinite. This terminology inspired by quantum theory reflects the fact the two pairs are dynamically strongly correlated.

Description of all pairs entangled to a fixed pair. — Let us fix a polynomial dynamical pair (P, a) parameterized by an algebraic curve C and for which the previous theorems apply (i.e. either the field of definition of the pair is a number field, or condition ( M ) holds). We would like now to determine all pairs that are entangled to (P, a).

In principle this problem is solvable by Ritt’s theory. Given a polynomial P , it is, however, very delicate to describe all polynomials Q for which (†) holds, in particular because there is no a priori bounds on the degrees of τ and π. Much progress have been made by Pakovich [117] but it remains unclear whether one can design an algorithm to solve this problem.

To get around this, we consider a more restrictive question which is to determine all pairs (P, b) that are entangled with (P, a). In this problem, the notion of symmetries of a polynomial plays a crucial role, and most of Chapter 3 is devoted to the study of this notion from the algebraic, topological and adelic perspectives. Suffice it to say here that the group Σ(P ) of symmetries of a complex polynomial P is the group of affine transformations preserving its Julia set. Of importance in the latter discussion is the subgroup Σ

(P ) of affine maps g ∈ Σ(P ) such that P

(g · z) = P

(z) for some n ∈ N

.

We also introduce the notion of primitive polynomials. A polynomial P is primitive if any equality P = g · Q

with g ∈ Σ(P ) implies n = 1.

These notions of symmetries and primitivity allow us to obtain the following neat statement.

Theorem D. — Let (P, a) be any active primitive non-integrable dynamical

pair parameterized by an algebraic curve defined over a field K of characteristic

0. Assume that K is a number field, or that ( M ) is satisfied.

10

INTRODUCTION

For any marked point b ∈ K[C] such that (P, b) is active, the following assertions are equivalent:

1. the set Preper(P, a) ∩ Preper(P, b) is infinite,

2. there exist g ∈ Σ(P ) and integers r, s ≥ 0 such that P

(b) = g · P

(a).

Note that this gives a positive answer to [80, Question 1.3] for polynomials.

Beware that the previous result does not quite describe the marked points parameterized by C which are entangled with (P, a). Indeed if s = 0 and r is sufficiently large, the solutions to the equation P

(b) = a are not necessarily regular functions on C: b belongs to a finite extension of K(C) of degree at most d

. In fact, we have:

Theorem E. — Let (P, a) be any active non-integrable primitive dynamical pair parameterized by an irreducible affine curve C defined over Q ¯ .

Then, any marked point b ∈ Q ¯ [C] that is entangled with a belongs to the set {g · P

(a) ; n ≥ 0 and g ∈ Σ

(P )} except for finitely many exceptions.

This result seems to be new even for the unicritical family.

It would obviously be more natural to assume the pair to be defined over an algebraically closed field of characteristic 0, but we use at a crucial step the assumption that (P, a) is defined over ¯ Q . Note that the theorem is not hard when ¯ Q is replaced by a number field.

Interestingly enough, the proof of this finiteness theorem relies on the same ingredients as Theorem C, namely the expansion of the B¨ ottcher coordinate, an algebraization result of adelic curves, and Ritt’s theory. The proof in fact shows that one may only suppose b ∈ Q ¯ (C).

Unicritical polynomials. — In the short Chapter 7, we discuss in more depth some aspects of unlikely intersection problems for unicritical polynomi- als.

Recall that in their seminal paper, Baker and DeMarco obtained the follow- ing striking result: for any d ≥ 2, and any a, b ∈ C , the pairs Preper(z

+ t, a), Preper(z

+ t, b) are entangled iff a

= b

. This result was further expanded by Ghioca, Hsia, and Tucker to more general families of polynomials and not necessarily constant marked points, see [80, Theorem 2.3].

Building on our previous results, we obtain the following statement.

Theorem F. — Let d, δ ≥ 2. If a, b are polynomials of the same degree and

such that Preper(z

+t, a)∩ Preper(z

+t, b) is infinite, then d = δ and a

= b

.

INTRODUCTION

After proving this theorem, we make some preliminary exploration of the set M of complex numbers λ ∈ C

such that the bifurcation locus ∂M

is connected, where we have set M

:= {t, λ

t ∈ K (z

+ t)}. We observe that λ ∈ M iff M

⊂ M(d, 0), and prove that M is a closed set of C

included in the unit disk, and containing the punctured disk D

(0, 1/8).

Special curves in the parameter space of polynomials. — We finally come back in Chapter 8 to our original objective, namely the classification of curves in Poly

which contain an infinite subset of PCF polynomials, and the proof of Baker and DeMarco’s conjecture claiming that these curves are cut out by critical relations.

A first answer to Baker and DeMarco’s question is given by the next result.

Theorem G. — Pick any non-isotrivial complex family P of polynomials of degree d ≥ 2 with marked critical points which is parameterized by an algebraic curve C containing infinitely many PCF parameters. Then we are in one of the following two situations.

1. There exists a symmetry σ ∈ Σ(P ) and a family of polynomials Q param- eterized by some finite ramified cover C

→ C such that P = σ · Q

for some n ≥ 2, and the set of t

∈ C

such that Q

is PCF is infinite.

2. There exists a non-empty subset A of the set of critical points of P such that for any pair c

, c

∈ A, there exists a symmetry σ ∈ Σ(P ) and integers n, m ≥ 0 such that

(2) P

(c

) = σ · P

(c

) ;

and for any c

∈ / A there exist integers n

> m

≥ 0 such that (3) P

(c

) = P

(c

) .

Following the terminology of [6, §1.4] inspired from arithmetic geometry, we call special any curve in Poly

containing infinitely many PCF polynomials.

Our theorem says that a special curve in the moduli space of polynomials of degree d either arises as the image under the composition map of a special curve in a lower degree moduli space, or is defined by critical relations (including symmetries) such that all active critical points are entangled.

This result opens up the possibility to give a combinatorial classification of

all special curves in the moduli space of polynomials of a fixed degree Poly

.

Recall that a combinatorial classification of PCF polynomials in terms of Hub-

bard trees has been developed by Douady and Hubbard [53, 54], Bielefeld-

Fisher-Hubbard [24] and further expanded by Poirier [120], and Kiwi [97]. We

12

INTRODUCTION

make here a first step towards the ambitious goal of classification of special curves using a combinatorial gadget: the critically marked dynamical graph .

We refer to §8.2 for a precise definition of critically marked dynamical graph.

It is a (possibly infinite) graph Γ(P ) with a dynamics that encodes precisely all dynamical critical relations (up to symmetry) of a given polynomial P . We show that to any irreducible curve C in the moduli space of (critically marked) polynomials, one can attach a marked dynamical graph Γ(C) such that Γ(P ) = Γ(C) for all but countably many P ∈ C. We then identify a class of marked graphs that we call special which arise from special curves.

Under the assumption that the special graph Γ has no symmetry and that its marked points are not periodic, we conversely prove that the set of polynomials such that Γ(P ) = Γ defines a (possibly reducible) special curve. Our precise statement is quite technical, see Theorem 51, but it provides an interesting correspondence between a wide family of special curves and combinatorial objects of finite type.

The proof of this result builds on two constructions of polynomials with special combinatorics, one by Bielefeld-Fisher-Hubbard of strictly PCF combi- natorics, and one by McMullen and DeMarco of dynamical trees [50]. Binding together these two results was quite challenging. We have been able to prove only a partial correspondence under simplifying additional assumptions.

Some technical details that we have worked out and hopefully clari- fied! — Beside presenting a set of new results, we have made special efforts to clarify some technical aspects of the standard approach to the unlikely in- tersection problem for polynomials. We emphasize some of them below.

– We include a self-contained proof (by J. Xie) of his algebraization result for adelic curves (Theorem 6).

– We give the full expansion of the B¨ ottcher coordinates for polynomials over a field of characteristic 0 without assuming it to be centered or monic (§2.4).

– We study over an arbitrary field the group of symmetries of a polynomial.

In particular, we give a purely algebraic characterization of this group (Theorem 15).

– We introduce the notion of primitivity in §3.4, which seems appropriate to exclude tricky examples of entangled pairs.

– We give a detailed proof that the height h

(t) = h

(a(t)) attached to

any polynomial dynamical pair is adelic (Proposition 4.15).

INTRODUCTION

– For a family of polynomials {P

} parameterized by an algebraic variety Λ, we consider the preperiodic locus in Λ × A

which is a union of count- ably many algebraic subvarieties. We study the set of points which are included in an infinite collection of irreducible components of the prepe- riodic locus (Theorem 12). This result is crucial to our specialization argument to obtain Theorem C and clarifies some arguments used in [85].

Open questions and perspectives. — There are many directions in which our results could find natural generalizations.

Let us indicate first why the restriction to families of polynomials is crucial for us. Given a family of rational maps R

parameterized by an algebraic curve C, and given any marked point a, then one can attach to the pair (R, a) a di- visor at infinity D

generalizing the definition (1) above. It is not completely clear, however, whether D

has rational coefficients. Some cases have been worked out by DeMarco and Ghioca [47] but the general case remains elusive.

One can next build a natural height by setting h

= h

(a(t)), but it is not completely clear if this height is a Weil height associated to D

(in the sense of Moriwaki [112]). There are instances (see [52]) where this height is not adelic. Although a notion of quasi-adelicity has been proposed by Mavraki and Ye [113] to get around this problem, a continuity of the metrizations in- volved in the definition of the canonical height (Theorem 1 in the polynomial case) turns out to be crucial for applications and remains completely open up to now. We also note that Ritt’s theory is much less powerful for rational maps leading to weaker classification of curves left invariant by product maps (see [118]). We also refer to [145] for a characterization of rational maps having the same maximal entropy measure; and to [111] for a version of Theorem B for constant families of rational maps (but varying marked point).

It would be extremely interesting to look at polynomial dynamical tuples parameterized by higher dimensional algebraic variety Λ and prove unlikely intersection statements. The obstacles to surmount are also formidable. It is already unclear whether the canonical height is a Weil height on a suitable compactification of Λ. Also in this case, the bifurcation measure is naturally defined as a Monge-Amp` ere measure of some psh function on Λ, and deal- ing with a non-linear operator makes things much more intricate. We refer, though, to the papers by Ghioca, Hsia & Tucker [81] and by Ghioca, Hsia

& Nguyen [79] for tentatives to handle higher dimensional parameter spaces

using one-dimensional slices.

14

INTRODUCTION

Let us list a couple of questions that are directly connected to our work.

(Q1) Prove the following purely archimedean rigidity statement: two complex polynomial dynamical pairs (P, a) and (Q, b) having identical bifurcation measures are necessarily entangled. One of the problem to get such a statement is to prove the multiplicative dependence of the degrees in this context. Observe that Theorem 38 yields this dependence but at the cost of a much stronger assumption.

(Q2) Is it possible to remove condition ( M ) and obtain Theorem C over any field of characteristic 0?

(Q3) Can one extend Theorem E to any field of characteristic 0?

(Q4) Give a classification of special (irreducible) curves in the moduli space of critically marked polynomials in terms of suitable combinatorial data.

Ideally, one would like to attach to each special irreducible curve a combi- natorial object (like a family of decorated graphs) and prove a one-to-one correspondence between special curves and these objects. It would also be interesting to study the distribution (as currents) of special curves whose associated combinatorics has complexity increasing to infinity.

Further more specific open problems can be found in the three sections §3.7 (related to Ritt’s theory), §5.6 (on extensions of Theorem B) and §8.8 (on special curves).

Acknowledgements. — We warmly thank Junyi Xie and Khoa Dang

Nguyen for sharing their thoughtful comments after reading a first version of

this work. We express our gratitude to Dragos Ghioca and Mattias Jonsson

for their very constructive remarks, and to Gabriel Vigny for many exchanges

on the rigidity of complex polynomials that greatly helped improving our

discussion on this matter.

NOTATIONS

Notations

– K ¯ the algebraic closure of a field K

– K

the ring of integers of a non-Archimedean metrized field – K ˜ the residue field of a non-Archimedean metrized field – C

the completion of an algebraic closure of K

– K a number field

– O(z) the set of Galois conjugates over an algebraic closure of K of a point z ∈ K

– M

the set of places of a number field, i.e. of norms extending the usual p-adic and real norms on Q

– K

the completion of K w.r.t v ∈ M

– C

the completion of an algebraic closure of K

– X an algebraic variety – O

the structure sheaf of X

– X

the analytification in the sense of Berkovich of X

– X

the analytification of X

when X is defined over a number field K and v ∈ M

– C an algebraic curve

– C ¯ a projective compactification of a curve C such that ¯ C \ C is smooth – C ˆ the normalization of ¯ C, and n : ˆ C → C ¯ the normalization map

– log

= max{0, log}

– D = {z ∈ C , |z| < 1} the complex open unit disk – S

the unit circle in C

– U the group of complex numbers of modulus 1

– U

= {z ∈ C , z

= 1} the group of complex m-th root of unity – U

the group of all roots of unity

– D

(z, r) = {|w − z| < r} the open disk of center z and radius r (as a subset of either K or the Berkovich affine line A

)

– D

(z, r) = {0 < |w − z| < r} the punctured open disk of center z and radius r (as a subset of either K or the Berkovich affine line A

) – D

(z, r) = {|w − z| ≤ r} the closed disk of center z and radius r (as a

subset of either K or the Berkovich affine line A

)

– D

(z, r) the open polydisk of center z and polyradius r = (r

, · · · , r

) (as a subset of either K

or the Berkovich affine space A

)

– D

(z, r) the closed polydisk of center z and polyradius r = (r

, · · · , r

)

(as a subset of either K

or the Berkovich affine space A

)

16

INTRODUCTION

– D

(z, r), D

(z, r), D

(z, r), D

(z, r) the respective open and closed disks in K

if v is a place on a number field K

– M

: the ring of analytic functions on the punctured unit disk D

(0, 1) that are meromorphic at 0

– C

(X) the space of compactly supported continuous functions on X – D(U ) the space of smooth (resp. model) functions on U

– ∆u the Laplacian of u – u, g subharmonic functions – h harmonic functions

– o(1), O(1): Landau notations

– g

the Green function associated to a polynomial P

– G(P ) the critical local height of a polynomial (the maximum of g

at all critical points)

– ϕ

the B¨ ottcher coordinate of a polynomial P at infinity – J(P ) the Julia set of a polynomial P

– K(P ) the filled-in Julia set of a polynomial P – µ

the equilibrium measure of a polynomial P – Crit(P ) the critical set of a polynomial P

– Σ(P ) the group of dynamical symmetries of a polynomial P

– Aut(P ) the group of affine transformations commuting with a polynomial P

– Aut(J ) the group of affine transformations fixing a compact subset J of the complex plane

– Preper(P, K) the set of preperiodic points lying in K of a polynomial P ∈ K[z]

– Poly

the space of polynomials of degree d

– Poly

the space of monic and centered polynomials of degree d – MPoly

the space of polynomials of degree d modulo conjugacy

– MPoly

the moduli space of critically marked polynomials of degree d modulo conjugacy

– MPair

the moduli space of dynamical pairs of degree d modulo conjugacy – Stab(P ) the stability locus of a holomorphic family of polynomials – T

the Chebyshev polynomial of degree d

– M

the monomial map of degree d

– (P, a) a dynamical pair (either holomorphic or algebraic)

NOTATIONS

– Bif(P, a) the bifurcation locus of a holomorphic polynomial pair

– g

= g

(a(t)) the Green function associated to a holomorphic polyno- mial pair

– µ

the bifurcation measure of a dynamical pair defined over a metrized field

– Preper(P, a) the set of parameters t such that the marked point a(t) is

preperiodic for P

CHAPTER 1

GEOMETRIC BACKGROUND

We briefly review some material from analytic and arithmetic geometry.

This includes the notion of subharmonic functions on analytic curves defined over a non-Archimedean or an Archimedean field; the construction of the Laplace operator on the space of subharmonic functions; a discussion of the notion of semi-positive and adelic metrics on a line bundle over a curve; and the definition of heights attached to an adelic metrized line bundle.

We then define various moduli spaces of polynomials of interest, and give a proof of an algebraization theorem of Xie [144] for germs of curves defined by adelic series.

This section will be mainly used as a reference for the rest of the book.

1.1. Analytic geometry

1.1.1. Analytic varieties

Let (K, | · |) be any complete metrized field. When the norm is non- Archimedean, we let K

= {|z| ≤ 1} be its ring of integers with maximal ideal K

= {|z| < 1}. We write ˜ K = K

/K

for its residue field, and

|K

| = {|z|, z ∈ K

} ⊂ R

for its value group.

If X = Spec(A) is an affine K-variety, we denote by X

its Berkovich analytification. As a set, it is given by the Berkovich spectrum of A, i.e. the set of all multiplicative semi-norms on A whose restriction to K is equal to the field norm. We endow it with the topology of the pointwise convergence for which it becomes a locally compact and locally arcwise connected space. The space X

is also endowed with a structural sheaf of analytic functions.

When K = C , we recover the complex analytification of X with its standard

euclidean topology and the structural sheaf is the sheaf of complex analytic

functions.

20

CHAPTER 1. GEOMETRIC BACKGROUND

When K is non-Archimedean, we refer to [20] for a proper definition of the structural sheaf. Any point x ∈ X

defines a semi-norm | · |

on A, and it is customary to write |f (x)| = |f |

for any f ∈ A. The kernel ker(x) is a prime ideal of A which may or may not be trivial. The quotient ring A/ ker(x) is a field and one denotes by H(x) its completion with respect to the norm induced by | · |

. It is the complete residue field of x. When H(x) is a finite extension of K, then we say that x is a rigid point.

One can naturally extend the analytification functor to the category of al- gebraic varieties over K using a standard patching procedure. In this context, the GAGA principle remains valid, see [20, Chapter 3].

1.1.2. The non-Archimedean affine and projective lines

Suppose (K, | · |) is a complete metrized non-Archimedean field. The analyti- fication of the affine line A

is the space of multiplicative semi-norms on K[T ] restricting to | · | on K. It is topologically an R -tree in the sense that it is uniquely pathwise connected, see [96, §2] for precise definitions. In particular for any pair of points x, y ∈ A

there is a well-defined segment [x, y].

We define closed balls as usual ¯ B(z

, r) = {z ∈ K, |z − z

| ≤ r}. To any closed ball is attached a point x

∈ A

defined by the relation |P (x

)| = sup

|P | for any P ∈ K [T ]. The Gauß point x

is the point associated to the closed unit ball x

= x

. It induces the Gauß norm on the ring of polynomials: |P (x

)| = max{|a

|} with P (T ) = P

a

T

.

When K is algebraically closed, points in A

fall into one of the following four categories. If the kernel of x is non-trivial, then x is rigid and x = x

for some z ∈ K. We also say that x is of type 1. When x = x

with r ∈ |K

| (resp. r / ∈ |K

|), we say that x is of type 2 (resp. 3). If x is not of one of the preceding types, then it is of type 4: one can then show that there exists a decreasing sequence of balls B

with empty intersection such that |P (x)| = lim

|P (x

)| for all P ∈ K[z], see [10, §1.2].

When K is not algebraically closed, and K

/K is any complete field exten- sion, the inclusion K[Z ] ⊂ K

[Z] yields by restriction a canonical surjective and continuous map π

: A

→ A

, and the Galois group Gal(K

/K) acts continuously on A

.

Let C

be the completion of an algebraic closure of K. Then A

is home- omorphic to the quotient of A

by Gal( C

/K), see [20, Corollary 1.3.6].

The group Gal( C

/K) preserves the types of points in A

, so that we may

define the type of a point x ∈ A

as the type of any of its preimage by π

in A

. Note that since the field extension C

/K is not algebraic in general,

1.1. ANALYTIC GEOMETRY

it may happen that some type 1 points in A

have trivial kernel hence are not rigid.

Any open subset of the affine line carries a canonical analytic structure in the sense of Berkovich. We shall refrain from defining this notion precisely and discuss only the case of balls and annuli.

The Berkovich analytic open unit ball D

(0, 1) is defined as the space of semi-norms | · |

∈ A

such that |x| := |T |

< 1. Its structure sheaf is the restriction of the analytic sheaf of A

. Any analytic isomorphism of D

(0, 1) is given by a power series of the form P

n≥0

a

T

with |a

| < 1, |a

| = 1, and

|a

| < 1 for all n ≥ 2.

For any ρ > 1, the standard open annulus A = A(ρ) of modulus ρ is the analytic subset of A

defined by {1 < |x| < e

}. Any analytic isomorphism of A is given by a Laurent series of the form P

n∈Z

a

T

with |a

| = 1 and

|a

|r > |a

|r

for all n 6= 1 and all 1 < r < e

, possibly composed with the inversion a/T with e

= |a|

.

The skeleton on A is the set of points Σ(A) = {x

, 0 < t < ρ}. Any automorphism of A leaves Σ(A) invariant so that the skeleton only depends on the analytic structure of A.

The projective line P

is homeomorphic to the one-point compactification of A

. The point at infinity in P

is rigid/of type 1.

1.1.3. Non-Archimedean Berkovich curves

Let (K, | · |) be any algebraically closed and complete non-Archimedean metrized field. Let C be any smooth connected projective curve defined over K. Berkovich [20, Chapter 4] proved that the geometry of the Berkovich analytification C

can be completely understood using the semi-stable re- duction theorem. Berkovich’s results were further expanded by Baker, Payne and Rabinoff in [7, 8]. We refer the interested reader to the unpublished monograph of Ducros [55] fo a detailed account on the geometry of any analytic Berkovich curve (not necessarily algebraic). Our presentation follows [7].

Models. — A model of C is a normal K

-scheme C that is projective and flat over Spec(K

), together with an isomorphism of its generic fiber with C. Denote by C

its special fiber: it is a proper scheme defined over the

(1)

The latter automorphism exists only if e

∈ |K

|

22

CHAPTER 1. GEOMETRIC BACKGROUND

residue field ˜ K . The valuative criterion of properness implies the existence of a reduction map red : C

→ C

which is anti-continuous, see [7, 1.3].

By a theorem of Berkovich [20, Proposition 2.4.4], the preimage by red of the generic point η

of any irreducible component E of C

is a single point in C

, which we denote by x

. Such a point is called of type 2. This terminology is compatible with the case C = P

. Indeed any point of the form red

(η

) ∈ P

is of type 2. Conversely, for any type 2 point x ∈ P

, there exists a model P of P

and a component E of P

such that x = x

.

If ¯ x is a closed point in C

, then red

(¯ x) is an open subset of C

whose boundary is finite and consists of those points x

where E is an irreducible component of the central fiber containing ¯ x, see [20, Theorem 4.3.1].

A model with simple normal singularities (or simply an snc model) is a smooth model for which the special fiber is a curve with only ordinary double point singularities, i.e. C admits a covering formally isomorphic to Spf(K

hx, yi/(xy − a)) for some a ∈ K

− {0} near any of its closed point, see [7, Proposition 4.3]. A fundamental theorem of Bosch and L¨ utkebohmert [28, Propositions 3.2 & 3.3] implies the following result.

Theorem 3. — 1. If x ¯ is a smooth point of C

lying in a component E, then red

(¯ x) is analytically isomorphic to the (Berkovich) open unit ball and its boundary in C

is equal to x

.

2. If x ¯ is an ordinary double singularity of C

and belongs to the two compo- nents E and E

, then red

(¯ x) is analytically isomorphic to a (Berkovich) open annulus of the form {1 < |z| < r} for some r ∈ |K

|, and its bound- ary is equal to {x

, x

}.

Skeleta. — The skeleton Σ(C) of an snc model is the union of all points x

for all components E of the special fiber, together with the union of all skeleta of the annuli red

(¯ x) for all singular points of C

. The skeleton contains no rigid points.

Since C

is a curve with only ordinary singularities, we may define its dual graph ∆(C) whose vertices (resp. of edges) are in bijection with the irreducible components of C

(resp. with the singular points of C

). The skeleton Σ(C) is a geometric realization of the graph ∆(C) in C

.

There a canonical continuous map τ

: C

→ Σ(C), [7, Definition 3.7].

For any irreducible component E of C

, τ

(x

) is equal to the vertex [E]

associated to E. More generally when red(x) is a smooth point lying in E

then τ

(x) = [E]. Finally when red(x) is the intersection of two components,

then τ

(x) belongs to the edge joining these two components.

1.1. ANALYTIC GEOMETRY

Since open subsets of the affine line are retractible, it follows that τ

is a retraction, and that C

is locally modeled on an R -tree. In particular, for any two points x 6= y there exists a continuous injective map γ : [0, 1] → C

such that γ(0) = x and γ(1) = y. Up to reparameterization, there are only finitely many such maps, and when C

is a tree this map is unique. The latter occurs iff the dual of some (or any) snc model of C is a graph having no loop.

Metrics. — Using the R -tree structure of the affine line, one can endow the complement of its set of rigid point H ( A

) = A

\ A

(K) with a complete metric d

as follows.

Suppose first that x

, x

∈ A

are associated to closed balls x

= x

as in the previous section. If B

and B

are not disjoint, then one is contained into the other say B

⊂ B

, and one sets d

(x

, x

) = log(diam(B

)/ diam(B

)).

When the balls are disjoint, we consider the closed ball of smallest radius B containing B

and B

, and set

d

(x

, x

) = log(diam(B )/ diam(B

)) + log(diam(B)/ diam(B

)).

It is a fact that this distance extends to H ( P

) = H ( A

) as a complete metric [10, §2.7].

Lemma 1.1. — Any injective analytic map from the open unit ball or from an annulus to the affine line induces an isometry for d

.

Sketch of proof. — Let us treat the case of the open unit ball and take any injective analytic map f : D

(0, 1) → A

. The image of f is an open ball, since it has at most one boundary point. Since any affine automorphism is an isometry for d

, we may suppose that f ( D

(0, 1)) = D

(0, 1). From the explicit description of the automorphism groups of the ball given in §1.1.2, we get that f(T ) = P

n

a

T

with |a

| < |a

| = 1 and |a

| < 1 for all n ≥ 2. The lemma then follows from the following estimation of the diameter of a ball:

diam(f (B(0, r))) = max

n≥1

|a

|r

. The case of the annulus is treated analogously.

Observe that the metric spaces ( D

(0, 1)∩ H ( A

), d

) and (A(ρ)∩ H ( A

), d

) are not complete. Their completions are respectively equal to ( D

(0, 1) ∪ {x

}, d

) and (A(ρ) ∪ {x

, x

}, d

).

Remark 1.2. — The preceding observation combined with Lemma 1.1 show

that D

(0, 1) ∩ H ( A

) ∪ {x

} and (A(ρ) ∩ H ( A

) ∪ {x

, x

}) are canonically

endowed with a complete metric that we shall again denote by d

.

24

CHAPTER 1. GEOMETRIC BACKGROUND

Let now C be any connected projective smooth curve, and denote by H (C) = C

\ C(K) the complement of the set of rigid points.

Proposition-Definition 1.3. — There exists a unique complete metric d

on H (C) such that the following holds. There exists > 0 such that any ana- lytic embedding f : U → C where U is either the open unit ball or an annulus of modulus ≤ induces an isometry from (U, d

) onto its image (f(U ), d

).

When the context is clear we shall drop the index C and simply write d

for the metric on H (C).

Sketch of proof. — Choose an snc model C. We shall build a metric d

on H (C) and latter show that this metric does not depend on the choice of the model.

By Bosch and L¨ utkebohmert’s theorem (Theorem 3) the Berkovich curve C

is the disjoint union of finitely many type 2 points (those of the form x

for some irreducible component E of the special fiber), of finitely many annuli A

(the preimages under the reduction map of the singular points of C

), and (possibly infinitely many) disjoints open balls B

.

Observe that the closures ¯ A

, B ¯

forms a compact covering of C

and that the intersection of any two of these pieces is either empty or equal to a point x

for some E as before. By Remark 1.2, we may endow each piece ¯ A

∩ H (C), B ¯

∩ H with a canonical metric d

for which they become an R -tree. We then extend the metric to H (C) by setting for any two points x, y ∈ H (C)

d

(x, y) = inf {Length(γ )}

where γ ranges over all injective continuous maps γ : [o, t] → H (C) such that γ(o) = x and γ(t) = y. Since each piece of the covering is an R -tree, one can take the infimum over paths which are parameterized by length, hence this infimum is taken over finitely many maps and is thus attained. The metric d

is complete since each piece is.

Let us now verify that d

satisfies the expected property that any embedding

of a ball or an annulus is isometric. Note that this will prove that distance

does actually not depend on the choice of the model. Take any embedding

of the open unit ball f : D

(0, 1) → C. For any s, t ∈ D

(0, 1)) note that

any path γ minimizing d

(f (s), f(t)) is actually included in f ( D

(0, 1)) since

the latter has only one boundary point. And it follows from Lemma 1.1 that

d

(f(s), f(t)) = d

(s, t) as required.

1.2. POTENTIAL THEORY

Let be half of the minimum of the length of loops included in the skeleton

∆(C). If f : A → C is an injective analytic map, and A is an annulus of mod- ulus ≤ , then again any path γ minimizing d

(f(s), f (t)) has to be included in A. We conclude the proof as before.

Remark 1.4. — Another construction of this metric is given in [7, §5.3] for details. We refer also to [55] for a construction of the metric in the case of an arbitrary analytic curve.

The metric topology ( H (C), d

) is stronger than the restriction of the com- pact topology C

to H (C). However for any snc model C, the restriction of both topologies to ∆(C) coincide.

1.2. Potential theory

1.2.1. Pluripotential theory on complex manifolds

We shall mainly use potential theory on Riemann surfaces. However we need to pass to higher dimensional complex varieties at a few places (most no- tably to prove Theorem 28 in Chapter 4). We review also basic properties of plurisubharmonic (psh) functions, see [92] for a general reference.

Let C be any Riemann surface. We let ∆ be the usual Laplace operator defined on C

functions in any holomorphic chart by ∆ϕ =

∂

∂

ϕ(z, z). This ¯ operator extends to any L

function in which case ∆ϕ is merely a distribution on C. Harmonic functions are those C

functions h : C → R such that ∆h = 0.

A subharmonic function u : C → R ∪{−∞} is a upper semicontinuous function such that for any harmonic function h on a holomorphic disk D ⊂ C such that u ≤ h on ∂D then u ≤ h on D. For any subharmonic function, ∆u is a positive measure. Conversely, any L

function whose Laplacian is a positive measure is equal a.e. to a (unique) subharmonic function, see [91, §1.6].

For any holomorphic function f : C → C , the function log |f| is subhar- monic. Subharmonic functions are stable by sum, multiplication by a positive constant, by taking maximum. Any decreasing limit of a sequence of subhar- monic functions remains subharmonic (or is identically −∞).

Let M be any connected complex manifold. To simplify the discussion we shall assume that the dimension of M is 2 (it covers all our needs for this book).

A function u : M → R ∪ {−∞} is said to be psh if it is upper semicontinuous and for any holomorphic map ı: D → M the function u ◦ ı is subharmonic.

Again for any holomorphic function f : M → C , the function log |f | is psh,

and psh functions are stable by sum, multiplication by a positive constant, by